Lie Symmetry Analysis and Dynamics of Exact Solutions of the (2+1)-Dimensional Nonlinear Sharma–Tasso–Olver Equation

In soliton theory, the dynamics of solitary wave solutions may play a crucial role in the fields of mathematical physics, plasma physics, biology, fluid dynamics, nonlinear optics, condensed matter physics, and many others. (e main concern of this present article is to obtain symmetry reductions and some new explicit exact solutions of the (2 + 1)-dimensional Sharma–Tasso–Olver (STO) equation by using the Lie symmetry analysis method. (e infinitesimals for the STO equation were achieved under the invariance criteria of Lie groups.(en, the two stages of symmetry reductions of the governing equation are obtained with the help of an optimal system. Meanwhile, this Lie symmetry method will reduce the STO equation into new partial differential equations (PDEs) which contain a lesser number of independent variables. Based on numerical simulation, the dynamical characteristics of the solitary wave solutions illustrate multiple-front wave profiles, solitary wave solutions, kink wave solitons, oscillating periodic solitons, and annihilation of parabolic wave structures via 3D plots.


Introduction
Most of the physical problems are inherently nonlinear because of their nature. ese types of nonlinear problems are of much interest to engineers, mathematicians, physicists, and many other researchers because of their physical applications in scientific areas such as engineering, biology, theoretical physics, plasma physics, and condensed matter physics (see [1]). To predict the possible behaviors of physical phenomena, exact solutions of these nonlinear PDEs are very important. e various types of rational solutions were investigated by analytical and numerical methods such as the Lie symmetry method [2,3], homogeneous balance method [4], tanh function method [5], Hirota bilinear method [6], Backlund transformation method [7], Darboux transformation method [8], the truncated Painleve expansion [9], and F-expansion method [10]. In solitary wave theory, a large variety of traveling waves are of much concern. Some important traveling waves are the kink waves and the solitons. e kink waves fall or rise from one asymptotic state to others, and the latter is the localized traveling waves which are asymptotically zero at large distances. Some potential applications of these wave solutions may be found in coastal areas and open oceans. e (1 + 1)-dimensional STO equation is of the form where α is some constant and the dissipative term u xxx provides damping of a small scale. A lot of researchers worked on the STO equation (1) due to its huge appearance in scientific applications. e STO equation is similar to the KdV equation and has been used to describe a wide range of physical phenomena of the evolution and interaction with the nonlinear waves, such as solitons and turbulence, fluid dynamics, continuum mechanics, and aerodynamics. e STO equation can also be used to describe how do nonlinear dispersive waves propagate in inhomogeneous media. It possesses the bi-Hamiltonian formulation and an infinite number of symmetries. Different nonlinear waves are described by the exact solutions for (1) with different forms. e generalized Kaup-Newell-type hierarchy of nonlinear evolution equations is related to the Sharma-Tasso-Olver equation from [11]. A large number of integrability properties of (1) are achieved in [12,13]. In [14], equation (1) was managed by Yan using the Cole-Hopf transformation method. By considering the improved tanh function method, fission and fusion for the Sharma-Tasso-Olver equation have some exact solutions in [15]. In ref. [16][17][18], some powerful techniques such as the Hirota direct method, extended hyperbolic function method, and Backlund transformation method are used for finding the explicit solutions of (1), respectively. Also, this equation was managed by using the tanh method and the extended tanh method (see [16]).
is research aims to find the exact-soliton solutions of (2 + 1)-dimensional STO equation. It is well known that the (2 + 1)-dimensional KP equation is an extended version of (1 + 1)-dimensional Korteweg-de Vries (KdV) equation. Analogously, (2 + 1)-dimensional STO equation is an extended version of (1 + 1)-dimensional STO equation. e extended form of (2 + 1)-dimensional STO equation is introduced in [19], which is accomplished by adding a new term u yy in (1 + 1)-dimensional STO equation (1) where α and β are two arbitrary constants and the unknown function u depends on the spatial variables x, y and temporal variable t. is equation encloses both nonlinear and linear terms. Using the tanh function method, the traveling wave solution of STO equation (2) is found as where c 5 and c 6 are arbitrary constants. Here, we will solve a new (2 + 1)-dimensional equation using the similarity transformation method. Felix Klein (1849-1925) and M. Sophus Lie (1842-1899) introduced this method to construct integration theory for ODEs. For further details, one can refer to textbooks [20,21] or some articles [3,[22][23][24][25][26][27] as well as references therein. During this process, some variable transformations take place, which is used to obtain new solutions from the existing ones using differential operators, called the generators of the symmetry group.
With the help of similarity forms, STM reduces the number of independent variables of the original PDE by one. As a result, the equation was reduced into a nonlinear PDE having one less independent variable. In the same way, one can reduce this nonlinear PDE into nonlinear ODE by reducing one independent variable, which can be solved mathematically.
Being motivated by the aforementioned references/ works, we would like to study to construct the explicit exactsoliton solutions by using the Lie symmetry approach. Besides, we derive various Lie symmetry reductions and localized solitary wave solutions for the considered equation.
Recently, Ben and Ma [19] investigated (2 + 1)-dimensional STO equation and found some rational solutions using the resulting trilinear form. Our analytic results in this paper are more general in terms of independent functions and entirely different from their works/findings. Furthermore, the physical behaviors of the soliton solutions are demonstrated through their evolution profiles. e remaining manuscript is structured as follows. In Section 1, we discuss various algorithms for solving the STO equation and a new form of the equation by adding one term u yy . e Lie symmetries, vector fields, infinitesimals, and an optimal system of one-dimensional subalgebra for the STO equation are depicted in Section 2. In Section 3, Lie symmetry reductions and analytical wave solutions are furnished. e dynamical analysis of attained soliton solutions is analyzed physically in Section 4. Lastly, a conclusion is drawn in Section 5.

Lie Symmetry Analysis
e motive of this present section is to describe the main steps used in obtaining the infinitesimal, Lie symmetries, and optimal system of one-dimensional subalgebra for STO equation (2). To find the invariant solutions of (2), the STM is applied. Since equation (2) is highly nonlinear, therefore to solve equation (2), we need to introduce one parameter (ε) Lie group of infinitesimals transformation as where the infinitesimals ξ 1 � ξ (1) , ξ 2 � ξ (2) , ξ 3 � τ, and η are the infinitesimals transformations for x, y, t, and u, respectively. e linked vector field V with the above group of transformation is defined as where the obtained functions ξ (1) , ξ (2) , τ, and η are found under the constraint: If equation (2) is symmetrical about the vector field (5), then V must fulfill the condition where Pr (4) V represents the fourth-order prolongation and is given by [21] 2 Mathematical Problems in Engineering Applying the above prolongation formula to (2) under invariance condition, the symmetry condition is where are the coefficients of Pr (4) V(Δ) � 0 and D x , D y , and D t are total derivative operators that can be expressed as in [28].
Using (10) and (9), then the determining equations On solving equation (12), we obtain infinitesimal generators as where c i ; 1 ≤ i ≤ 4 are arbitrary constants and f 1 (t) and f 2 (t) are functions obtained in the processing of infinitesimals. e choice of f i ′ s can achieve some physical structures of solutions of equation (2). e symmetry algebra of equation (2) can be obtained by the given vectors It has been noted from the commutative relations of vector fields that the Lie group of transformations formed an infinite-dimensional Lie algebra with functions f 1 (t) and f 2 (t) and its commutation table (Table 1) is skew-symmetric with each diagonal entry zero.
Mathematical Problems in Engineering e Lie series to calculate the adjoint relation is furnished as Furthermore, an optimal system of one-dimensional subalgebra is achieved by means of Olver's technique [21]. e desired optimal system for STO equation (2) is obtained by using adjoint table (Table 2). Following is the given onedimensional subalgebras for STO equation (2):

Symmetry Reductions and Explicit Solutions
In this section, we will construct some exact-soliton solutions with the aid of 1-dimensional optimal system of symmetry subalgebras. To achieve exact-invariant solutions for the considered equation (2), the corresponding Lagrange's system is chosen as where u(x, y, t) � U(X, T) and X � x − y, T � t.
Since equation (18) is also a nonlinear PDE and contains one dependent and two independent variables, we will again apply STM on this equation to construct new infinitesimal generators of the form where k 1 and k 2 are arbitrary constants and g 1 (T) is an arbitrary function. en, the required characteristic equation is On solving equation (20), we get the following similarity form where X 1 � X − (k 2 /k 1 )T. By using (21) into (18), then reduced form of ODE is which on integration gives where C is constant of integration and ′ represents the derivative of U 1 w.r.t X 1 . One of the particular solutions of equation (23) is where Hence, the exact explicit solution of equation (2) is given as  (2) is converted into where Again applying STM on equation (26), obtained form of infinitesimals is where k 3 , k 4 , k 5 , and k 6 are constants. en, the associated Lagrange's system is found as On solving equation (28), we get a new similarity form of U as with X 2 � X − (k 4 /k 3 )Y as a similarity variable. By using equation (29) into (26), we get an ODE of the form Since equation (30) is tedious nonlinear ODE, therefore by taking k 5 � 0 in equation (30), we attain one particular solution of the type where b 0 and b 1 are arbitrary constants. us, the explicit exact solution for STO equation (2) is 3.3. Subalgebra T 3 � V 4 + V 5 � (z/zx) + yf 1 (t)(z/zu).

If a 2 ≠ 0 in Equation (43).
en, the characteristic equation is as follows: where A 1 � (a 1 /a 2 ) and A 3 � (a 3 /a 2 ). Integration of equation (44) yields the following form of U where X 3 � Y. Using (45) into (42), we get an ODE as e solution of ODE (46) is given as where A and B are integrating constants. On comparing equations (40), (45), and (47), the explicit solution for equation (2) is obtained as Taking the appropriate choice of A � 1 and B � 2 ����������� � (c 2 /(2A 1 c 3 β)) in equation (47), we get us, from equations (40), (45), and (49), the leading solution is of the form erefore, the similarity form is where X 4 � Y and U 4 (X 4 ) are the similarity variable and similarity function, respectively. Substituting equation (52) in (42), one gets the following ODE: 6 Mathematical Problems in Engineering (53) On solving equation (53), where C and D are called integrating constants. From (40), (52), and (54), we will obtain the analytic solution of the STO equation Solution of (56) predicts the similarity function with similarity variables X � x − (c 4 t/c 3 ) and Y � y − (c 2 t/c 3 ). Incorporating equation (57) into equation (2) yields a PDE of the form By applying the STM on (58), we get a new set of infinitesimals as where a i ; 1 ≤ i ≤ 4 are arbitrary constants. e Lagrange form for (59) reads 3.6.1. If a 3 ≠ 0 in Equation (59). en, we have dX where B 1 � (a 1 /a 3 ), B 2 � (a 2 /a 3 ), and B 3 � (a 4 /a 3 ). To get the solution of (2), the new similarity form of (61) is with X 5 � X − (B 2 Y/B 1 ) as the similarity variable. Using the value of (62) into (58), we get an ODE of the form Integration of (63) provides where D is constant of integration. It is clearly seen that equation (64) is difficult to solve.

3.6.2.
If a 3 � 0 in Equation (59). en, given characteristic equation is It follows the similarity form as where X 6 � X − (a 2 Y/a 1 ). From equations (66) and (58), we obtain the ODE as By setting the particular values of arbitrary constants as a 1 � 1, a 2 � 1, c 1 � 1, c 2 � 1, c 3 � 1, α � 1, and β � 1, the reduced form of ODE is which on integration gives Mathematical Problems in Engineering 7 where C is constant of integration. One of the particular solutions of equation (69) is found as where A 0 is an arbitrary constant. us, the explicit exact solution of (2) is given as 3.6.3. If a 3 � 0 and a 1 � 0. en, Lagrange equation is which on solving proposes the following form of U: where X 7 � Y. Substituting the value of (73) into (58), we get ODE as e solution of this ODE is where α 0 and α 1 are arbitrary constants. us, from equations (75), (73), and (57), the analytic exact solution of (2) can be furnished as      u(x, y, t) � α 1 + − α 0 a 2 c 2 + a 4 c 4 t + a 4 c 3 x a 2 c 3

Analysis and Discussion
In the present section, a discussion about graphical structures of solutions is carried out. Since mathematical expressions do not produce the wave propagation of flow, we need physical as well as dynamical analysis of these solutions. Here, explicit analytic solutions of the STO equation (2) has been given by equations (3)  is profile is traced at t � 1, for − 20 ≤ x ≤ 20, − 20 ≤ y ≤ 20. e kink solution is a localized solution, i.e., u tends to a finite number when t ⟶ ∞. It was observed by numerical simulation that the behavior of exiting wave depends on time variation. It turns into single solitons at t � 11 and is then annihilated after t � 16. Figure 2 demonstrates periodic solitary wave profiles of u given by equation (35). e choice of arbitrary functions is taken as f 1 (t) � tan(t), h 1 (t) � cot(b 0 t + b 1 ), h 2 (t) � sin(t), and arbitrary constants as b 0 � 5, b 1 � 10, and β � 4 for space range − 25 ≤ x ≤ 25, − 25 ≤ t ≤ 25. In Figure 2(b), the corresponding 3D plot is shown for y � 6. It was observed that periodicity decreases as the value of y increases. is nonlinear behavior of soliton solution generally reveals the intersection of Riemann wave and long wave. Figure 3 interprets the multiple-front solitary wave solution profile of u given by equation (38) at y � 2, for range − 25 ≤ x ≤ 25, − 25 ≤ t ≤ 25. e choice of arbitrary functions is taken as h 1 (t) � cos(t), h 2 (t) � (tan(t 2 + t + 1)/(2t + 1)), f 2 (t) � sin(t), and parameter β � − 0.5. Initially, wave profile reveals curve-front wave nature of solution which gets converted into plane-front wave as y increases. Figure 4 shows the parabolic profile for the solution given by equation (48). Solution profile is observed at t � 2, for − 15 ≤ x ≤ 15, − 40 ≤ y ≤ 40, and the values of arbitrary constants are taken as A 1 � 0.110, A 3 � 0.115, c 1 � 2, c 2 � 0.001, c 3 � 1, c 4 � − 0.12, A � 3, B � 1, β � 1, b 0 � 10, and b 1 � 0.91, and the choice of arbitrary functions is taken as f 1 (t) � b 0 t + b 1 and f 2 (t) � ((2b 0 t + b 1 )/(tan(b 0 t 2 + b 1 t + 5))). e nonlinearity of solution (48) is annihilated at t � 20. is parabolic profile of solution can be utilized in making the design of ballistic missiles. Figure 5 is plotted between spatial and temporal axes x − t which represent the asymptotic wave profile for solution (50) and is illustrated at y � 1, for − 15 ≤ x ≤ 15, − 40 ≤ t ≤ 40. e values of constants are taken as A 1 � 0.110, A 3 � 1.019, c 1 � 2, c 2 � 0.001, c 3 � 1, c 4 � 0.12, β � 1, b 0 � 10, and b 1 � 0.91, and those of functions are taken as f 1 (t) � ((2b 0 t + b 1 )/(b 0 t 2 + b 1 t + 5)) and f 2 (t) � tanh(t).

Conclusion
In this research work, we have obtained some closed-form explicit solutions for the (2 + 1)-dimensional Sharma-Tasso-Olver (STO) equation by means of Lie approach, which are given by equations (3), (25), (32), (35), (38), (48), (50), (55), (71), and (76). ese achieved solutions are entirely distinct from the earlier findings [19]. Also, these reported results are revealed to be more general because it contains arbitrary constants and independent functions f 1 (t), f 2 (t), h 1 (t), and h 2 (t). All the generated results are analyzed physically through Figures 1, 2, 3, 4, and 5 which shows parabolic, periodic solitary waves, oscillating traveling waves, kink wave profiles, etc. e dynamics of solitary wave profiles are vividly exhibited via Mathematica. is research shows that the Lie transformation method can be extended for obtaining explicit exact-analytic solutions of many other NLEEs that are emerging in optical physics, plasma physics, chemical physics, acoustics, etc. In summary, the proficient Lie symmetry technique that we implemented is robust, competent, and efficient to solve analytically.

Data Availability
All data used to support the findings of this study are included within the article.